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\title[Solution of a Problem in Health Care Finance Risk Management]{Solution of a Problem in Health Care Finance Risk Management}
\address{{\lsuper a}1711 NW 55th Terrace\ Gainesville, FL 32605 \\ USA } %required
\email{nurse.statistician@yahoo.com}  %optional


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%% the abstract has to preceed the command \maketitle:

\keywords{Risk, Risk theory, Professional Caregiver Insurance Risk}

\begin{abstract}

Professional Caregiver Insurance Risk (PCIR) refers to the transfer of health insurance risks to health care providers.
These risk transfers occur in global capitation, health provider and managed care organization and insurer contracting and in the Medicare and Medicaid Prospective Payment Systems. 

PCIR turns the benefits achieved by efficient insurance transfers, from individuals to risk aggregating insurers, and eliminates their benefits by inefficiently disaggregating them by passing small subsets of the insurance risks to health care providers. PCIR turns health care providers into tiny, extraordinarily inefficient insurers. As a consequence, health providers, as inefficient insurers, are less likely to meet profit goals, more likely to incur high losses and must slash patient benefits to maintain their liquidity.

\end{abstract}
\maketitle
%% start the paper here:

\begin{document}
\section{Introduction}\label{S:one}

	Arrow et. al. (1) suggested: ``... different payment mechanisms, such as bundled or global payments and capitation...'' first, among their  recommendations, for building more efficient health care (finance) systems. Similar insurance risk transfers occur in the Prospective Payment Systems, Diagnosis Related Groups, Global payment programs, Episode based payment programs. We will refer to all insurance risk transfer mechanisms generically as "Professional Caregiver Insurance Risk" (PCIR) but readers are encouraged to consider global capitation, the most familiar as the paradigm insurance risk transfer. 
	
	Risk managers have too long failed to address whether PCIR is efficient enough to accomplish this goal. If not, the increased use of PCIR, bundled payments and similar risk transferring mechanisms, will further compromise our health care (finance) systems. Efforts to compare PCIR and fee for service models are difficult because they rarely exist apart in otherwise identical settings. Confounding variables in different sites and spillover effects in the same sites, leave most research inconclusive at best.(2-6) Understanding the true nature of the risks their facilities should be a focal concern for all risk managers.
	
	This paper starts ``up-river,'' addressing the inefficiency of PCIR by comparing the operating characteristics of small and large insurers. PCIR usually involves large, risk transferring entities (e.g. Managed Care Organization, Insurer, Medicare, or Medicaid) passing insurance risks to many smaller, insurance risk assuming, provider organizations, such as a physician practices, hospitals, long term care facilities or home health agencies.(7-15) Hence, the viability of PCIR hinges on whether large and small insurers are equally profitable, subject to equal risks of operating losses, and equal probabilities of insolvency. For PCIR to be viable approach, small insurers must: Be no less efficient than large insurers, No less profitable, Incur operating losses no more often, and have no higher risks of insolvency than large insurers. We shall show that this is not the case.
	
	The harms PCIR causes are always most severe at the level of a single provider and a single financial evaluation period. The smaller the insurer/provider or the shorter the evaluation period, the more inefficient the insurer/provider becomes. Small insurers operating results are more variable than large insurers, so during each financial period some insurers or providers will have marked deviations from average costs. Some insurers and providers, when evaluated on the basis of their most recent costs, will look very bad, despite the fact that they are efficient caregivers and provide high quality services or care. 
	
	We can compare the performance of large and small insurers using statistical sampling theory, risk theory and financial analysis.(16-23) This paper demonstrates that PCIR mechanisms are too inefficient to extend and that risk managers should be alerting their organizations to the risks they present.

\section{Background}\label{S:Background}

	PCIR advocates assume that health care providers are wasteful and inefficient, providing more services than needed to generate more income. PCIR advocates believe that fee for service providers over-test, over-diagnose, and over-treat patients and implementing PCIR encourages providers to become less inefficient. PCIR critics fear that risk assuming providers will under-test, under-diagnose, and under-treat patients.(2-6,24)
Unethical, profit maximizing, providers exist in both systems, maximizing net income by delivering excessive care under fee for service and inadequate care under PCIR. These unethical providers harm patients by exposing them to excessive testing and intervention or by failing to diagnose and treat them as early as they could.

	There is also widespread misunderstanding about insurance. Jonas (24) (See footnote pg 9) suggests that health insurance is not ``real insurance'' because people will use all their benefits eventually. Correcting misconceptions about insurance and PCIR is difficult because few people recognize that the most important consideration is insurer size. With regard to Jonas' concern, we note that ``whole life'' insurance always pays full benefits but it is the timing of premiums and benefits that determines feasibility.(19,16,17,25)

	Risk managers, health policy analysts and providers need to understand insurance rate making and operating results. This paper begins building these capabilities. We begin by avoiding faulty assumptions about waste and inefficiency that hide the inefficiency of PCIR, concentrating instead; on the effect insurer size has on insurer's operating results. 	

	We can demonstrate the flaws in PCIR with a simple model and familiar statistical tools if we assume that the health care (finance) systems are already efficient. We then analyze how portfolio size affects operating results for large and small insurers (small risk assuming health care providers), immediately before and after the implementation of PCIR.

	The key to insurance is the variation in insurer's loss ratios as functions of insurer portfolio size.(10,17,18,26,27) Many risk managers assume that because both large and small insurers have identical ``expected loss ratios'' when selecting policyholders at random, they should have identical probabilities of outcomes at loss ratios other than the expected values. To demonstrate that this is not true we make modest assumptions about a large ``Paradigm Insurer'' (PI), using the Central Limit Theorem to make inferences about other insurers.(28) 

	We cannot account for all the risk insurers' face. Instead, we focus on the routine variation in insurer's fortunes. A pandemic would sap the strength of any health insurer, and most would not survive even a fairly modest epidemic. Our model focuses on insurer's non-cataclysmic risk exposure.

\section{Statistical sampling theory}\label{S:StatisticalTheory}

	In statistical sampling theory we refer to populations, samples, and measurable, quantitative variables (28). Our population is all possible health insurance policyholders. Our samples are ``portfolios'' of randomly selected policyholders, usually comprised of policyholders insured by a single insurer. Our ``variable'' is the loss ratio (21,27,29), the ratio of aggregated losses (health care costs) to aggregated premiums for individual policyholders, portfolios, and the entire population.

	Policyholders pay \$4,000 in premiums and have unlimited potential losses, though losses higher than \$250,000 are extremely rare. Policyholders with no health care costs have loss ratios of 0.0000 (\$0/\$4,000), losses of \$5,000 generate loss ratios of 1.2500 (\$5,000.00/\$4,000) and a loss of \$250,000 generates a loss ratio of 62.5000 (\$250,000/\$4,000).
	
	If a risk manager knows nothing about the loss ratio for a population s(he) might randomly select an individual population member, and use their loss ratio, as an estimate of the Population Loss Ratio (PLR). This random selection is unlikely to equal the PLR. The risk manager can get more accurate estimates of the PLR by selecting random portfolios of ``n'' policyholders, calculating the portfolio average loss ratios, calling them ``Population Loss Ratio Estimates'' (PLREs), and use the PLRE to estimate the PLR. This produces PLR estimates that tend to lie closer to the PLR than individual policyholder's estimates. However, even these portfolio PLREs fall around the population loss ratio (PLR) in ways predicted by the Central Limit Theorem (CLT).(28) The CLT states that increasing portfolio size will increase PLRE accuracy. Insurers increase the accuracy of their PLREs by writing more policies. We note that if insurers do not randomly select policyholders, systematically insuring only the lowest (highest) risk policyholders, their premium revenues would be excessive (inadequate). 
	
	The standard deviation of the loss ratio, for a population of policyholders, measures the variation in loss ratios for individual policyholders. The standard error plays the same role for PLR estimates (PLREs) derived from portfolios. The standard error for a portfolio of size ``n'' is the standard deviation of the population of all possible portfolio loss ratios derived from random selections of portfolios of size ``n'' selected from the same population of potential policyholders. (28) 
	
	If an insurer could repeatedly select and service policyholders, calculating its PLRE each time, the standard error describes the probability distribution for a single random selection from the population of all PLREs of size ``n.'' 	The CLT provides a formula for calculating portfolio standard errors if we know the standard deviation for the population of individual loss ratios, or if we know the value of the standard error for a specific portfolio size ``n.'' 	

If the standard deviation for an individual loss ratio is s, the standard error for the PLRE for a portfolio of size ``n,'' is $$s_n = \frac{s}{\sqrt{n}}$$. The standard deviation (standard error) of the loss ratio, for a population (portfolio) of policyholders, measures the distance, from the PLR, that random selections of individual loss ratios (portfolio PLREs) lie. The CLT states that for populations that are approximately ``normally distributed,'' a condition we assume to be satisfied because insurer's portfolios are quite large, the probability that a randomly selected single (portfolio) loss ratio will lie within 1, 2 or 3 standard deviations (standard errors) of the PLR is about 0.6827; 0.9545 and 0.9973, respectively. 

\section{Insurer Operating Results}\label{S:OperatingResults}
	
	Insurer's loss ratios are important because they determine whether insurers make generous profits, modest profits, break even, lose money, or become insolvent (bankrupt). We can break each dollar of insurer's premiums into ratios of specific costs per dollar of premium. We will assume that the ratio of the largest cost, losses, is on average, equal to the population loss ratio (PLR), 0.7500. All insurers incur non-loss ``operating expenses'' of \$0.15 per dollar of premium, assessing ``profit provisions'' and ``risk premiums'' of \$0.05 per dollar of premium for their insurance services. 
	
	All insurers have \$0.85 per dollar of premium to pay policyholder benefits, without incurring net operating losses. The risk premium, a hedge against higher than average losses, protects the insurer's profit provision. As a last resort, insurers can tap their profit provisions, using these funds to pay benefits and still avoid net operating losses. Beyond PLREs of 0.8500, insurers require other funds to cover their obligations or they become insolvent.
	
	Insurers earn profits of at least 10\% (\$0.10 per dollar of premium) when their PLREs are no higher than 0.7500. They earn profits of at least 5\% (\$0.05 per dollar of premium) when their PLREs are no higher than 0.8000. They break even when their PLREs equal 0.8500 and they incur losses when their PLREs exceed 0.8500. Insurer's losses will be higher than 5\% when their PLREs exceed 0.9000 and we assume that they will incur substantial, solvency threatening, losses (10\% or more), when their PLREs exceed 0.9500.

\section{Paradigm Insurer}\label{S:ParadigmInsurer}
	
	The Paradigm Insurer (PI), randomly selects 1,000,000 policyholders (Loss ratios), collects premiums, pays losses and, at year end, calculates its estimate (PLRE) of the population loss ratio (PLR). We assume that PI's standard error, $s_{1,000,000}$, is 0.05 and will show below that we can calculate the value for the standard error for any portfolio size using the CLT (28) and $s_{1,000,000}$.
	
	We will use portfolio adjusted standard errors and our assumption that PLREs are normally distributed, to analyze and compare the performance of PI and larger and smaller insurers, before and after PI shifts from fee for service to PCIR. We summarize PI's performance in Exhibit 1 Column 4 but in more detail, PI:

\begin{itemize}
\item Earns premiums of \$4,000,000,000 (\$4,000 per year, per policyholder)
\item Has underwriting and non-loss related expenses of \$600,000,000 (Expense ratio = 0.15)
\item Has a profit Provision = \$200,000,000 (0.05 * \$4,000,000,000)
\item Risk Premium = \$200,000,000 (0.05 * \$4,000,000,000)
\item P[Losses $\leq$ \$3,000,000,000 = P[Profit $\geq$ 10\%] = 0.5000
\item P[Losses $\leq$ \$3,200,000,000 = P[Profit $\geq$ 5\%] = 0.8413
\item P[Losses $\leq$ \$3,400,000,000 = P[Avoid losses] = 0.9772
\item P[Losses $\geq$ \$3,600,000,000 = P[Avoid losses $\geq$ 5\%] = 0.9987
\item P[Losses $\geq$ \$3,800,000,000 = P[Avoid losses $\geq$ 10\%] = 1.0000
\end{itemize}	
	
	The probabilities we have assigned to PI's outcomes are calculated by evaluating the cumulative probabilities for PI's normally distributed PLREs (N[0.75, 0.05]), for loss ratios at, or below, 0, 1, 2, 3, and 4 standard errors above the PLR. These evaluation points are equivalent to PLREs of: 0.7500; 0.8000; 0.8500; 0.9000 and 0.9500, respectively. 

\section{Standard Errors By Portfolio Size}\label{S:StandardErrors}

	When insurers A, B, PI, D and E randomly select portfolios of: 307,000,000; 10,000,000; 1,000,000; 100,000 and 10,000 policyholders from the same population, they select PLREs from different, normally distributed, populations. Each insurer's distribution is centered at the PLR, but the standard errors for portfolios of size "n," are calculated using our assumption that s1,000,000 = 0.05 and this formula: 
	
\begin{equation}
s_n = s_{1,000,000} * \frac{1,000,000}{\sqrt{n}}	
\end{equation}
	
	The greater accuracy of larger insurer's PLREs, and the lower accuracy of smaller insurer's PLREs, are reflected in the sizes of their standard errors: 0.0029; 0.0158; 0.0500; 0.1581 and 0.5000, respectively, as shown in Exhibit 1 Row 4.

Although the loss ratios we used earlier were 0, 1, 2, 3 and 4 standard errors above the PLR for PI, these same loss ratio evaluation points will be higher (lower) numbers, of standard error units, above the PLR for insurers larger (smaller) than PI, affecting their respective cumulative probabilities and the probabilities assigned to specific operating results. Larger insurers have more probability of loss ratios below evaluation points higher than the PLR, and small insurers have more probability of loss ratios above all evaluation points higher than the PLR. As a result, larger insurers are more likely to earn profits and avoid losses than smaller insurers, despite the fact that all insurers select individual policyholders, at random, from the same population.



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